Solved Examples and Worksheet for Identifying and Using the Slopes of Parallel and Perpendicular Lines

Q1Find the slope of the line parallel to - 2x + 8y = 10.
A. 15
B. 2
C. 8
D. 14

Solutions
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Step: 1
Two lines are said to be parallel if they have same slope and different y-intercepts.
Step: 2
The slope intercept form of the equation - 2x + 8y = 10 is y = 14x + 54.
Step: 3
The slope of the line - 2x + 8y = 10 is 14.
Step: 4
So, the slope of the line parallel to - 2x + 8y = 10 is 14.
Correct Answer is :   14
Q2Which is an equation of a line perpendicular to the line y = - 14x + 18?

A. y = - 45x + 13
B. y = 4x + 37
C. y = - 43x + 7
D. y = 34x + 1

Solutions
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Step: 1
y = - 14x + 18
  [Write the equation of the line.]
Step: 2
Slope of the line is - 14.
  [Find the slope of the line using y = mx + b.]
Step: 3
If m is the slope of the line perpendicular to the line y = - 14x + 18 , then
m × - 14 = - 1 ⇒ m = 4
Step: 4
EquationSlope
y = - 43x + 7
y = 34x + 1
y = - 45x + 13
y = 4x + 37		- 43
34
- 45
4
  [Find the slope of the line using y = mx + b.]
Step: 5
So, the line y = 4x + 37 is perpendicular to the line y = - 14x + 18.
Correct Answer is :   y = 4x + 37
Q3The two lines y = 3x + 4 and y = - 13x + 34 are ___________.
A. parallel to each other
B. perpendicular to each other
C. passes through origin
D. vertical

Solutions
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Step: 1
Slope of the line y = 3x + 4 is 3.
  [Compare with y = mx + b.]
Step: 2
Slope of the line y = - 13x + 34 is - 13.
  [Compare with y = mx + b.]
Step: 3
Product of the slopes of two lines = 3 × (- 13) = - 1
  [Multiply.]
Step: 4
So, the lines y = 3x + 4 and y = - 13x + 34 are perpendicular to each other.
  [Two lines are perpendicular, if the product of their slopes is - 1.]
Correct Answer is :   perpendicular to each other
Q4Choose the parallel lines from the set of lines:
4x + 5y + 4 = 0, 5x + 6y + 8 = 0, 4x + 5y + 6 = 0, 5x + 9y + 8 = 0

A. 5x + 6y + 8 = 0 and 5x + 9y + 8 = 0
B. 4x + 5y + 4 = 0 and 5x + 6y + 8 = 0
C. 4x + 5y + 4 = 0 and 4x + 5y + 6 = 0
D. 5x + 9y + 8 = 0 and 4x + 5y + 4 = 0

Solutions
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Step: 1
Slope of the line 4x + 5y + 4 = 0 is - 45.
  [Slope = -xcoefycoef.]
Step: 2
Slope of the line 5x + 6y + 8 = 0 is - 56.
  [Slope = -xcoefycoef.]
Step: 3
Slope of the line 4x + 5y + 6 = 0 is - 45.
  [Slope = -xcoefycoef.]
Step: 4
Slope of the line 5x + 9y + 8 = 0 is - 59.
  [Slope = -xcoefycoef.]
Step: 5
Lines 4x + 5y + 4 = 0 and 4x + 5y + 6 = 0 are parallel.
  [Slopes of parallel lines are equal.]
Correct Answer is :    4x + 5y + 4 = 0 and 4x + 5y + 6 = 0
Q5Find the value of k if the line ky = (k -1)x + 3k is parallel to y = 2x + 5.

A. - 3
B. 3
C. - 1
D. 2

Solutions
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Step: 1
Slope of y = 2x + 5 is 2.
Step: 2
Slope of ky = (k -1)x + 3k is k  -1k
  [Compare with y = 2x + 5.]
Step: 3
Slopes of two nonvertical lines are equal.
Step: 4
Since the two lines are parallel, so 2 = k -1k
Step: 5
2k = k - 1⇒k = - 1
Correct Answer is :   - 1
Q6If 7x + 13y = - 6 and x + jy = 5 are parallel, then find the value of j.


A. - 713
B. 713
C. 137
D. - 137

Solutions
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Step: 1
7x + 13y = - 6
  [First equation.]
Step: 2
y = - 7x13 - 613
  [Write it in slope-intercept form.]
Step: 3
Hence, the slope is - 713.
Step: 4
x + jy = 5
  [Second equation.]
Step: 5
y = - xj + 5j
  [Write it in slope-intercept form.]
Step: 6
Hence, the slope is - 1j.
Step: 7
Since the lines are parallel, - 1j = - 713
Step: 8
So, j = 137.
Correct Answer is :    137
Q7Which among the following are the slopes of two perpendicular lines?
A. 4 and - 14
B. - 4 and - 14
C. 4 and 14
D. 4 and - 4

Solutions
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Step: 1
Two lines are perpendicular, if the product of their slopes is - 1.
Step: 2
For the slopes 4, - 14
4 × - 14 = - 1
  [Multiply and simplify.]
Step: 3
For the slopes - 4, - 14
- 4 × - 14 = 1
  [Multiply and simplify.]
Step: 4
For the slopes 4, 14
4 × 14 = 1
  [Multiply and simplify.]
Step: 5
For the slopes 4, - 4
4 × - 4 = - 16
  [Multiply and simplify.]
Step: 6
So, 4 and - 14 are the slopes of two perpendicular lines.
Correct Answer is :   4 and - 14
Q8Which of the following lines are parallel?

A. x - 3y + 11 = 0, -x + 3y -7 = 0
B. x + 3y + 11 = 0, -x + 3y -7 = 0
C. x + 5y + 11 = 0, -x + 3y -7 = 0
D. x + 3y + 11 = 0, x + 3y -7 = 0

Solutions
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Step: 1
Two lines which have the same slope and different y-intercepts are called parallel lines.
Step: 2
Write all the sets of equations in the standard slope-intercept form, y = mx + b and verify.
Step: 3
For the lines x - 3y + 11 = 0, -x + 3y -7 = 0:
Step: 4
x - 3y + 11 = 0 ⇒y = x3 + 113
Step: 5
Slope = 13 and y-intercept = 113.
  [Compare with y = mx + b.]
Step: 6
-x + 3y -7 = 0 ⇒ y = x3 + 73
Step: 7
Slope = 13 and y - intercept = 73
  [Compare with y = mx + b.]
Step: 8
Slopes are same and y-intercepts are different.
Step: 9
So, x - 3y + 11 = 0, -x + 3y -7 = 0 are parallel.
Correct Answer is :   x - 3y + 11 = 0, -x + 3y -7 = 0
Q9Which of the following pairs of lines are parallel?

A. -x + 53y + 3 = 0, x - 53 y - 4 = 0
B. -x + 35 y + 3 = 0, x - 53 y - 4 = 0
C. -x + 53y + 4 = 0, -x - 53 y - 3 = 0
D. x + 53 y + 3 = 0, x - 53 y + 4 = 0

Solutions
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Step: 1
Two lines which have the same slope and different y-intercepts are called parallel lines.
Step: 2
Write all the sets of equations in the standard slope-intercept form, y = mx + b and verify.
Step: 3
For the lines -x + 53y + 3 = 0, x - 53y - 4 = 0:
Step: 4
-x + 53 y + 3 = 0 ⇒ y = 35 x - 9/5
Step: 5
Slope = 35and y-intercept = -95
  [Compare with y = mx + b.]
Step: 6
x - 53y - 4 = 0 ⇒ y = 35 x - 125
Step: 7
Slope = 35 and y-intercept = - 125
  [Compare with y = mx + b.]
Step: 8
Slopes are same and y-intercepts are different.
Step: 9
So, -x + 5/3 y + 3 = 0, x - 5/3 y - 4 = 0 are parallel.
Correct Answer is :   -x + 53y + 3 = 0, x - 53 y - 4 = 0
Q10Choose the parallel lines from the set of lines:
2x + 5y + 3 = 0, 3x + 4y + 3 = 0, 5x + 4y - 7 = 0, 2x + 5y - 7 = 0
A. 2x + 5y + 3 = 0 and 3x + 4y + 3 = 0
B. 3x + 4y + 3 = 0 and 5x + 4y - 7 = 0
C. 5x + 4y - 7 = 0 and 2x + 5y - 7 = 0
D. 2x + 5y + 3 = 0 and 2x + 5y - 7 = 0

Solutions
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Step: 1
Two lines which have the same slope and different y-intercepts are called parallel lines.
Step: 2
check slopes for all the lines given.
Step: 3
Slope of the line 2x + 5y + 3 = 0 is - 25.
  [Slope = -xcoef/ycoef.]
Step: 4
Slope of the line 3x + 4y + 3 = 0 is - 3/4.
  [Slope = -xcoef/ycoef.]
Step: 5
Slope of the line 5x + 4y - 7 = 0 is - 54.
  [Slope = -xcoef/ycoef.]
Step: 6
Slope of the line 2x + 5y - 7 = 0 is - 2/5.
  [Slope = -xcoef/ycoef.]
Step: 7
Lines 2x + 5y + 3 = 0 and 2x + 5y - 7 = 0 are parallel.
  [Slopes are equal for both the lines.]
Correct Answer is :   2x + 5y + 3 = 0 and 2x + 5y - 7 = 0
Q11If 3x + 7y + 2 = 0 and 7x - ky + 5 = 0 are perpendicular, then find the value of k.

A. 53
B. 3
C. 37
D. 5

Solutions
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Step: 1
3x + 7y + 2 = 0
  [First equation.]
Step: 2
y = - 37x - 27
  [Write it in slope-intercept form.]
Step: 3
Hence, the slope is - 37.
Step: 4
7x - ky + 5 = 0
  [Second equation.]
Step: 5
y = 7k x + 5k
  [Write it in slope-intercept form.]
Step: 6
Hence, the slope is 7k .
Step: 7
Since the lines are perpendicular, (- 37) × (7k) = - 1
  [Product of slopes = - 1 for two perpendicular lines.]
Step: 8
So, k = 3.
  [Simplify.]
Correct Answer is :   3

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